Quantile Regression: An Exploration of the 2023 Behavioral Risk Factor Screening Survey Data

Authors: Hamilton Hewlett & Laura Sikes, MPH, CPH
Group: Not Sure
Instructor: Dr. Samantha Seals
Date: November 19, 2024

Introduction: Regression Overview

• Regression is a method used to predict outcomes based on the behavior of variables within a dataset relative to a mean.
• Of an infinite number of lines that can be drawn through the data, the one best describing an overall relationship is the one with the smallest sum of the square of the errors.
• The errors measure the distance from each data point to the line, then each value is squared, and those squares are summed. 
• The distances are squared in order to eliminate the negative properties of the distances that are found between the line and the data points below it. 
• The line of best fit is one that produces the smallest sum of these squares. 
• The formula for the regression line is

\[ \hat{y} = \beta_0 + \beta_1 x \]

Introduction: Regression Overview

In the regssion line equation \[ \hat{y} = \beta_0 + \beta_1 x \] where the intercept \(\beta_0\) is \[ \beta_0 = \frac{\sum y}{n} - \beta_1 \frac{\sum x}{n} \] and where the slope \(\beta_1\) is \[ \beta_1 = \frac{n \sum x y - \sum x \sum y}{n \sum x^2 - \left(\sum x\right)^2} \]

Based on the above, predictions can be made about what would happen along the regression line where there are no data points.

Introduction: Quantile Regression

• Quantile regression (QR) is a statistical approach that divides a distribution into different sections and finds the median point inside each one. 
• This approach is especially useful when a distribution contains exceedingly high or low values, given that medians are not as influenced by extreme values as compared to means. 
• QR is often an excellent solution when only one upper or lower end of a distribution contains a key variable. 
• Variables may enjoy unique relationships within different sections of a distribution (Buhai, 2005). 
• The equation for QR is

\[ Q_Y(\tau \mid X) = X \beta(\tau) \]

• Q is the quantile function, τ represents the quantile of interest, Y is the response variable given the X predictor variable(s), β is the regression coefficient (Koenker, 2005).

Introduction: Quantile Regression

• QR is a non-parametric method, meaning no assumptions are made about the sample size or the normality of the distribution (Cohen et al., 2016).
• Koenker and Machado (1999) identified the absence of a suitable goodness-of-fit assessment for this statistical method. 
• The authors developed a method with inspiration from R².  R² is the coefficient of determination, which identifies the difference between the regression line and the mean. 
• This coefficient’s values lie between 0 and 1, with values approaching 1 indicating ever-greater portions of the variation are explained by the relationship among the variables.

Introduction: Quantile Regression, Goodness of Fit

The formula for R² is as follows:

\[ R^2 = 1 - \frac{\sum \left(y_i - \hat{y}\right)^2}{\sum \left(y_i - \bar{y}\right)^2} \] - The sum of observed values \(y_i\)  is subtracted by the sum of fitted values \(\hat{y}\) and squared, forming the numerator. 
- The sum of observed values \(y_i\) minus the sum of sample mean \(\bar{y}\) and then squared, forming the denominator.
- This formula may also be represented as \[ R^2 = 1 - \frac{RSS}{TSS} \] where RSS is the residual sum of squares and TSS is the total sum of squares.
- The post-hoc testing methods Koenker and Machado proposed are Δ_n(τ), W_n(τ), and T_n(τ), which the authors claimed would vastly enhance the capacity for quantile regression inference. 
- The theory behind these methods is beyond the scope of this paper.

Introduction: Quantile Regression Applications

• Applications for QR are found in a diverse array of fields, including economics, ecology, real estate, and healthcare.
• Olsen et al. (2017), employed QR in a retrospective cohort study using data from insurer claims to identify factors affecting costs of infections following inpatient surgical procedures.
• Serious infections were categorized as those occurring in hospital or needing follow-up surgery, and were found to be the most costly.
• QR provided a method to focus on different factors associated with costs for the serious surgical infections located within that quantile. 

Introduction: Quantile Regression Applications

• Waldmann (2018) utilized body mass index (BMI) data to illustrate the usefulness of QR.
• The author’s goal was to answer questions about the outer portions of the distribution, which in this case was boys with the highest BMI.
• QR afforded the author insight into how predictor variables can have stronger or weaker effects within different areas of the distribution. 

• Cohen et al. (2016) used QR to evaluate the relationships between community demographics and perceived resilience. 
• The authors divided the distribution into percentiles and quartiles, and found that significant relationships existed inside several quantiles that were not significant when using linear regression on the entire distribution. 

Introduction: Quantile Regression Applications

• Staffa et al. (2019) employed QR to understand relationships between ventilator dependence and hospital length of stay, suspecting that these relationships changed across different quantiles. -The authors found stronger relationships between the predictor variables inside the upper quantiles of the length of stay outcome. 

• Machado and Silva (2005) explored the use of QR for count data by incorporating artificial smoothness, allowing inferences to be made about the conditional quartiles.  The authors note that count data is often severely skewed, making it an excellent candidate for QR which is more resilient against skewness given its use of medians over means.

Introduction: Quantile Regression Applications

Methodology: Study Design and Data

• Cross-sectional study using 2023 Behavioral Risk Factor Surveillance System (BRFSS) published by Centers for Disease Control and Prevention (CDC).
• BRFSS is a yearly survey conducted via land-line and cellular telephone (CDC, 2024).
• Includes U.S. residents aged 18 years and older.
• Survey topics include demographic information, socioeconomic factors, physical and mental health status, chronic conditions, and behaviors affecting physical and mental health.

Methodology: Predictor Variables - Hewlett Project

Methodology: Response Variable - Hewlett Project

Methodology: Analytical Methods - Hewlett Project

Methodology: Response Variable - Sikes Project

• Days per month a respondent’s poor health affected his or her activities of daily living (ADL)
• Respondents asked to quantify the number of days per month having poor mental or physical health interfered with their ability to conduct ADLs.
• Available responses were numeric values 1 – 30, none, refused, or don’t know/not sure.
• The authors note that quantile regression is a non-parametric method, meaning no assumptions are made about the sample size or the normality of the distribution.

Methodology: Predictor Variables - Sikes Project

• Weight: Available responses including numeric values 50 – 776 in pounds, don’t know/not sure, refused, and a separate indicator for weight measured in kilograms.
• Education: Available responses included never attended school or only kindergarten, grades 1 through 8, grades 9 through 11, grade 12 or GED, 1 to 3 years of college, 4 years of college or more, and refused.
• Income in U.S. dollars: Available responses included <10,000, $10,000 to < $15,000, $15,000 to < $20,000, $20,000 to < $25,000, $25,000 to < $35,000, $35,000 to < $50,000, $50,000 to < $75,000, $75,000 to < $100,000, $100,000 to < $150,000, $150,000 to < $200,000, $200,000 or more, don’t know/not sure, and refused.
• Physical health: Number of days during the previous 30 days in which the respondant’s physical or mental health were not good, with available responses including numeric values 1 - 30, refused, or don’t know / not sure.
• Sex: Available responses were limited to male or female.

Methodology: Operationalization of Variables - Sikes Project

• Dataset filtered to include only residents of the state of Florida and removed of values containing “Refused,” “Not sure / Don’t know”, or missing.
• Response variable ADL was filtered to include only numeric responses, and recoded to four groups as follows: “1-7 Days”, “8-14 Days,” “15-21 Days”, and “>21 Days”.
• Education was recoded as follows: “Less than High School,”, “High School or GED,” “Some College”, and “4 Years of College or Higher.
• Income was recoded into six groups as follows: “<$25,000”, “$25,000-$34,999”, “$35,000-$49,000”, “$50,000-$74,999”, “$75,000-$99,999”, and “>$100,000”.
• Physical health and mental health were each recoded into four groups as follows: “1-7 Days”, “8-14 Days,” “15-21 Days”, and “>21 Days”.
• All statistical analyses were performed using RStudio version 2024.04.2, build 764, “Chocolate Cosmos” release for Windows (Posit Software, PBC, 2024).

Methodology: Analytical Methods - Sikes Project

• QR was performed to explore ADL as a function of weight.
• The distribution was divided into quartiles (\(\tau\) = 0.25, 0.5, 0.75) to find the median values within each.
• Ordinary least squares regression (OLS) was performed using the same variables.

Results: Descriptive Statistics - Sikes Project

• Majority (42.6%) of considered respondents reported between 1 to 7 days of poor mental or physical health negatively impacting ADL, followed by >21 days at 26.3% (Table 1a - 1c).
• Nearly one-third (31.9%) reported having greater than 21 days of poor physical health per month, while a similar portion (30.2%) reported having over 21 days of poor physical health.
• Over one-half (50.2%) reported annual income of under $34,000, with just 13.1% reporting an income of over $100,000.
• Weight ranged from 82 to 500 pounds with a median of 180 (SD = 53.58).
• This sample is heavily skewed female with only 39.1% (n=493) identifying as male.

Results: Descripive Statistics - Sikes Project

Table 1a.

Results: Descripive Statistics - Sikes Project

Table 1b.

Results: Descripive Statistics - Sikes Project

Table 1c.

Results: Descripive Statistics - Sikes Project

Table 1d.

Results: Quantile Regression - Sikes Project

Table 2.

Note. \(\alpha\) = 0.05

Results: Quantile Regression - Sikes Project

Figure 1.

Days per Month ADLs Affected by Poor Mental or Physical Health as a Function of Weight (lbs)

Results: Hewlett Project

Discussion

• QR produced outputs describing the relationships between our predictor variables and our response variables at different segments of the distributions.
  

Discussion: Sikes Project

Table 2.

• At the first quartile (\(\tau\) = 0.25), a coefficient of 0.005 (95% CI = -0.002, 0.012) indicates that for each one pound increase in weight, the number of days per month of ADL affected by poor mental or physical health increases by 0.005 days at the \(\alpha\) = 0.95 level (p=0.158). This finding is not statistically significant.
• At the second quartile (\(\tau\) = 0.50), a coefficient of 0.000 (95% CI = -0.013, 0.013) indicates that for each one pound increase in weight, the number of days per month of ADL affected by poor mental or physical health is not detected at the \(\alpha\) = 0.95 level (p=1.000). This finding is not statistically significant.
• At the third quartile (\(\tau\) = 0.75), a coefficient of 0.032 (95% CI = -0.004, 0.068) indicates that for each one pound increase in weight, the number of days per month of ADL affected by poor mental or physical health increases by 0.032 days at the \(\alpha\) = 0.05 level (p=0.078). This finding is not statistically significant.

Discussion: Sikes Project

• Since the third quartile (\(\tau\) = 0.75) was marginally but not statistically significant with a p-value of 0.078 at the \(\alpha\) = 0.05 level, it may be useful to investigate these relationships further.
  
• Weight (lbs) may have a larger effect at the higher extreme of the distribution not examined in this study.
• Future studies could examine the effect at the ninetieth (\(\tau\) = 0.90) and ninety-fifth (\(\tau\) = 0.95) percentiles.

Conclusion

• QR is well-represented within the literature and has numerous advantages over traditional regression models when relationships between variables are suspected to change across quantiles.
• QR is an excellent analytical method to employ when stronger relationships are suspected at the extremes of the distribution.
  
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• With applications from real estate to life-and-death healthcare outcomes, QR affords a unique perspective into the nuances of our data.

References

WILL BE COMBINED AND UPDATED

Centers for Disease Control and Prevention (CDC). (2024). 2023 BRFSS Survey Data and Documentation. Retrieved October 12, 2024 from https://www.cdc.gov/brfss/annual_data/annual_2023.html

Posit Software, PBC. (2024). RStudio desktop. Retrieved October 13, 2024 from https://posit.co/download/rstudio-desktop/